What if a body loses 86,6% of its mass as couples of gamma photons?

[Alfredo Tifi]

TL;DR Summary

Does the experiment of Einstein's 1905, in which he derives E = mc², lead to an absurd?

After emitting two photons (or any other kind of energy) in the direction of motion and in the opposite direction, the velocity of the body (a big charged and unstable particle) remains unchanged, while the kinetic energy decreases. This entails a decrement of the rest mass, or of the inertia of that body.
We could measure the decrement of inertia, if we want, by accelerating the "big particle" in the same accelerating system (e.g. two electrically charged plates) before and after the emission of a couple of photons, and find that the Δv in the second -same- repeated accelerating device is increased as a consequence of its loss of inertia. We also could decelerate the big particle, finding that |Δv| is greater after the double emission. But we don't do that. We leave the big particle going on undisturbed emitting its couples of photons and remaining at the same (relatively low) velocity v.
Perhaps, once the same body has emitted many couples of gamma photons, eventually, its rest mass could approach zero! That would mean it could tend to zero rest mass with a relatively low velocity. This appears quite weird to me.

 

[Ibix]

Whether the mass could approach zero or not would depend on what it was made of, I think. For example a hot body radiating heat wouldn't fall below its cold rest mass (there is no way for it to emit more energy), whereas matter/anti-matter annihilating could fall to zero mass.

I don't see what's bothering you, I guess. The process, at least conceptually, isn't radically different from evaporation which is a perfectly common phenomenon. Both carry away mass by "emitting" stuff. The new (in 1905) fact is that the emitted stuff can be light.

 

[Dale]

Summary: Does the experiment of Einstein's 1905, in which he derives E = mc², lead to an absurd?

Perhaps, once the same body has emitted many couples of gamma photons, eventually, its rest mass could approach zero! That would mean it could tend to zero rest mass with a relatively low velocity. This appears quite weird to me.

A body composed of an electron and a positron achieves this routinely. Not only is it theoretically possible it is used on a daily basis across the world for medical imaging.

What is your concern?

 

[Alfredo Tifi]

Replying to Ibix and Dale.
You should consider this as a "gedankenexperiment", as was the one of Einstein.
Then, what is strange, or paradoxical, is what concerns the remaining energy-mass, which is "slow".

Let's take another, simpler, case. The body has a dynamite charge which explodes sending 49.999999999999% mass due East and 49.999999999999% mass due West, while the ε = 0.000000000002 % remaining has the energy-mass of two gamma photons plus the rest energy-mass of a neutrino. The residual tiny body ε is still in the same reference frame of the initial body, because of the perfect symmetry of the explosion. Now observe the whole phenomena from the point of view of the laboratory. They observe a small nanobody moving due North at the same velocity of a slug, in the vacuum, vₛ, carrying two molecules of TNT with it. After the explosion, there remains a positronium atom with the same velocity vₛ and direction N.
A spectrometer reveals the fragments of the explosion and calculates that the positronium still has unchanged velocity vₛ →N. Now, the positronium atom annihilates into two gamma photons and a neutrino remains which has velocity vₛ.

So we remain with a super tiny residual inertial mass which tends to zero while its velocity does not tend to light speed if observed from the laboratory reference system and still from their own RS. This is strange. We never observe slug-neutrinos and either still neutrinos. We know that as the rest mass of a body approaches zero its velocity should approach c from every reference system. In a gedankenexperiment, I can imagine the transformation from a massive particle to a zero rest mass particle as gradually as I want. The interpretative system should remain coherent.

 

[Nugatory]

We know that as the rest mass of a body approaches zero its velocity should approach c from every reference system.

That is not correct, and this mistaken belief may be the source of your confusion. No matter how small the rest mass, as long as it is non-zero there will be a frame in which the speed and momentum are zero, and the problem is easiest to visualize in that frame.

For definiteness, we can consider the body to be made up of positronium - "atoms" made up of bound electron-positron pairs. There are no external forces acting on the body, so if we start with the frame in which the body is at rest it will remain at rest in that frame. The mass of the body goes down as the electrons and positrons annihilate and photon pairs are emitted, but it remains at rest. Eventually the mass goes to zero when the last positronium atom is gone - but now we don't have a body with zero rest mass that is constrained to move at speed $c$, we have nothing.

@Ibix made an analogy with evaporation, and it's a good one.

 

[Ibix]

As long as the mass of an object is non-zero, the idea of it being stationary is fine. It doesn't happen with neutrinos in practice because the tiniest disturbance is enough to kick a neutrino up to 99-point-whatever percent of light speed relative to whatever's trying to detect it, but there's nothing wrong with this in principle. And when you've reduced your mass to zero, there's nothing there so speed isn't defined.

 

[Dale]

We know that as the rest mass of a body approaches zero its velocity should approach c from every reference system.

This is not correct in general. If it has non zero mass, no matter how small, there is always a frame where it is at rest.

If you fix the energy and reduce the mass then the velocity does approach c. However, that is not in every reference frame, it is in the reference frame where the energy is fixed. Remember energy is frame variant.

If you reduce the mass and reduce the energy then you can get the behavior you describe.

 

[Alfredo Tifi]

Thank you all for your thoughtful rebuttals, which eventually convicted me of the non-contradictory nature of a small velocity for the tiniest body or even at rest (zero velocity) unless the body is not completely converted and disappears, together with the hypothetical contradiction.

I have now something to learn from Dale. That there are reference frames where the energy is fixed while the rest mass can diminish. I suppose this is the result of taking into account the complete equation for the squared momentum quadrivector, of which E² = (mc²)² is a sub-case. And I presume that the kinetic energy of a body, as evaluated from the laboratory system for the moving body (energy is frame variant!) is included in the total energy (which could be kept fixed).

All the reflections of mine, came out from adopting the same question of Jim Baggott in "Mass": what is inertia, or rest-mass? Is that a sort of "illusion"? I am not sure, once this post will be closed, that I will be closer to be able to formulate an answer.

 

[Ibix]

That there are reference frames where the energy is fixed while the rest mass can diminish. I suppose this is the result of taking into account the complete equation for the squared momentum quadrivector, of which E² = (mc²)² is a sub-case.

More or less. He's simply proposing a non-inertial frame where he measures the kinetic energy increasing to compensate for the mass decreasing. So you pick a (non-inertial) frame where the velocity of the object increases as its mass decreases such that $\gamma mc^2$ is constant. You can do the same in non-relativistic physics if you want.

what is inertia, or rest-mass?

The modulus of the object's four-momentum. That's all relativity says about it.

 

[Dale]

He's simply proposing a non-inertial frame where he measures the kinetic energy increasing to compensate for the mass decreasing.

Oh, that would work, but I wasn’t thinking so clever. I was actually thinking about a single inertial frame and a bunch of particles where all of the particles had the same energy but different masses. I didn’t write it well.